Polynomial Division Calculator
Use this polynomial division calculator to divide one polynomial by another, find the quotient, find the remainder, and see the division identity. Enter a dividend such as \(2x^3+3x^2-5x+6\) and a divisor such as \(x-2\), then the calculator will show the quotient, remainder, rational form, and step-by-step polynomial long division work.
Polynomial division is the algebra process used to divide a polynomial expression by another polynomial expression. It works like numerical long division, but instead of place values, you compare leading terms and subtract polynomial products. The result is written as a quotient plus a remainder over the divisor.
Divide polynomials
Type each polynomial using powers like x^2, x^3, and regular plus or minus signs. The divisor cannot be zero. Use one variable only.
What is polynomial division?
Polynomial division is the process of dividing one polynomial by another polynomial. The polynomial being divided is called the dividend, and the polynomial you divide by is called the divisor. The answer is called the quotient, and any part left over is called the remainder. This is similar to ordinary number division. For example, when \(17\) is divided by \(5\), the quotient is \(3\) and the remainder is \(2\), because \(17=5(3)+2\). Polynomial division follows the same structure, but the objects being divided are algebraic expressions.
In polynomial division, the key identity is \(P(x)=D(x)Q(x)+R(x)\). Here, \(P(x)\) is the dividend, \(D(x)\) is the divisor, \(Q(x)\) is the quotient, and \(R(x)\) is the remainder. The remainder must have a lower degree than the divisor. If the remainder is zero, the divisor divides the dividend exactly. In that case, the divisor is a factor of the dividend.
For example, if \(x^3-8\) is divided by \(x-2\), the quotient is \(x^2+2x+4\) and the remainder is \(0\). This means \(x^3-8=(x-2)(x^2+2x+4)\). Since the remainder is zero, \(x-2\) is a factor of \(x^3-8\). This example also connects polynomial division to factoring special products such as the difference of cubes.
Main polynomial division identity
\[ P(x)=D(x)Q(x)+R(x) \]
Where:
- \(P(x)\) is the dividend.
- \(D(x)\) is the divisor.
- \(Q(x)\) is the quotient.
- \(R(x)\) is the remainder.
- The degree of \(R(x)\) must be less than the degree of \(D(x)\).
How to use the polynomial division calculator
Enter the dividend polynomial in the first box and the divisor polynomial in the second box. Use the caret symbol for powers, such as \(x^2\), \(x^3\), or \(x^4\). You can use positive terms, negative terms, constants, missing powers, and decimal coefficients. For example, \(2x^4-x^3+6x-5\) is valid even though it does not include an \(x^2\)-term. Missing terms are treated as coefficients of zero.
The variable box lets you choose the variable letter. The default is \(x\). If your problem uses \(t\), \(y\), or \(m\), change the variable to that letter. Use one variable only. This calculator is designed for single-variable polynomial division, which is the standard form used in algebra, precalculus, and many calculus preparation topics.
After you click the calculate button, the tool parses both polynomials, compares leading terms, divides those leading terms, multiplies the divisor by the new quotient term, subtracts, and repeats the process until the remainder has a lower degree than the divisor. The final result is shown as a quotient, a remainder, and a rational expression of the form \(Q(x)+\frac{R(x)}{D(x)}\).
Input tip: Write powers as \(x^2\), \(x^3\), and \(x^4\). Write \(x\) instead of \(1x\) if you want, and write negative terms normally, such as \(3x^2-5x+7\).
Polynomial long division method
Polynomial long division is the most general school method for dividing polynomials. It works for divisors of degree \(1\), degree \(2\), degree \(3\), or higher. The method begins by comparing the leading term of the dividend with the leading term of the divisor. The leading term is the term with the highest power of the variable. Dividing the leading term of the current dividend by the leading term of the divisor gives the next term of the quotient.
After finding a quotient term, multiply the entire divisor by that quotient term. Then subtract the product from the current dividend. The result becomes the new working polynomial. Repeat the process until the degree of the remaining polynomial is lower than the degree of the divisor. At that point, the remaining polynomial is the remainder.
This process can feel long at first, but it is systematic. Each step reduces the degree of the working polynomial, which means the process must eventually stop. The calculator follows this same algorithm. It is not just simplifying the final expression; it is applying the division algorithm step by step.
Leading-term step
\[ \text{Next quotient term} = \frac{\text{leading term of current dividend}}{\text{leading term of divisor}} \]
For example:
\[ \frac{2x^3}{x}=2x^2 \]
Step-by-step process
- Write the dividend and divisor in standard form, from highest degree to lowest degree.
- Include zero placeholders for missing powers if doing long division by hand.
- Divide the leading term of the current dividend by the leading term of the divisor.
- Write the result as the next term of the quotient.
- Multiply the entire divisor by that quotient term.
- Subtract the product from the current dividend.
- Bring down or continue with the remaining terms.
- Repeat until the remainder has a degree lower than the divisor.
- Write the final answer as quotient plus remainder over divisor.
Worked example: divide \(2x^3+3x^2-5x+6\) by \(x-2\)
Start with the dividend \(2x^3+3x^2-5x+6\) and the divisor \(x-2\). The leading term of the dividend is \(2x^3\), and the leading term of the divisor is \(x\). Divide \(2x^3\) by \(x\) to get \(2x^2\). This is the first term of the quotient.
\[ \frac{2x^3}{x}=2x^2 \] Multiply: \[ 2x^2(x-2)=2x^3-4x^2 \] Subtract: \[ (2x^3+3x^2-5x+6)-(2x^3-4x^2)=7x^2-5x+6 \]
Now repeat the process. Divide \(7x^2\) by \(x\) to get \(7x\). Multiply \(7x(x-2)=7x^2-14x\), then subtract.
\[ \frac{7x^2}{x}=7x \] \[ 7x(x-2)=7x^2-14x \] \[ (7x^2-5x+6)-(7x^2-14x)=9x+6 \]
Now divide \(9x\) by \(x\) to get \(9\). Multiply \(9(x-2)=9x-18\), then subtract.
\[ \frac{9x}{x}=9 \] \[ 9(x-2)=9x-18 \] \[ (9x+6)-(9x-18)=24 \]
The quotient is \(2x^2+7x+9\), and the remainder is \(24\). Therefore:
\[ \frac{2x^3+3x^2-5x+6}{x-2} = 2x^2+7x+9+\frac{24}{x-2} \] or \[ 2x^3+3x^2-5x+6=(x-2)(2x^2+7x+9)+24 \]
Quotient and remainder explained
The quotient is the main polynomial result of the division. The remainder is what is left after the divisor can no longer divide the remaining polynomial. In number division, \(17\div5=3\) remainder \(2\). In polynomial division, the remainder may be a constant, a linear expression, or another lower-degree polynomial. The only rule is that the degree of the remainder must be lower than the degree of the divisor.
If you divide by a linear polynomial such as \(x-2\), the remainder must be a constant because constants have degree \(0\), which is lower than degree \(1\). If you divide by a quadratic polynomial such as \(x^2+1\), the remainder can be linear, such as \(3x-5\), because degree \(1\) is lower than degree \(2\). This degree rule tells you when to stop the long division process.
The quotient and remainder are not separate guesses. They must satisfy the polynomial division identity. If the dividend is \(P(x)\), the divisor is \(D(x)\), the quotient is \(Q(x)\), and the remainder is \(R(x)\), then the identity \(P(x)=D(x)Q(x)+R(x)\) must be true. You can always check your answer by multiplying the divisor by the quotient and then adding the remainder.
Writing the answer as a rational expression
Polynomial division often appears inside rational expressions. If \(P(x)\) is divided by \(D(x)\), the result can be written as \(Q(x)+\frac{R(x)}{D(x)}\). This is useful when simplifying algebraic fractions, analyzing slant asymptotes, decomposing rational functions, or rewriting expressions in a more useful form.
For example, if dividing \(2x^3+3x^2-5x+6\) by \(x-2\) gives quotient \(2x^2+7x+9\) and remainder \(24\), then the rational expression form is:
\[ \frac{2x^3+3x^2-5x+6}{x-2} = 2x^2+7x+9+\frac{24}{x-2} \]
This form shows the polynomial part and the leftover fractional part. If the remainder is zero, the fractional part disappears, and the division is exact. If the remainder is not zero, the fraction remains part of the final expression.
Polynomial long division vs synthetic division
Polynomial long division works for any polynomial divisor. Synthetic division is a faster shortcut, but it works only in specific cases, usually when dividing by a linear divisor of the form \(x-c\). For example, dividing by \(x-2\) can be done with synthetic division using \(2\). Dividing by \(x+3\) can be done with synthetic division using \(-3\). But dividing by \(x^2+1\) requires long division or another more general method.
Synthetic division is efficient because it uses only coefficients, but it can hide the reasoning from students who are still learning. Polynomial long division shows why each quotient term appears and how the subtraction step works. For learning, long division is often the better foundation. Once you understand long division, synthetic division becomes easier to use correctly.
| Method | Best for | Works with | Main limitation |
|---|---|---|---|
| Polynomial long division | General polynomial division | Linear, quadratic, cubic, or higher-degree divisors | More writing and more steps |
| Synthetic division | Quick division by \(x-c\) | Usually linear divisors with leading coefficient \(1\) | Not general enough for every divisor |
| Factoring | Exact simplification | Expressions with common or recognizable factors | Only works easily when factors are visible |
Remainder theorem and factor theorem
The remainder theorem says that when a polynomial \(P(x)\) is divided by \(x-c\), the remainder is \(P(c)\). This is a powerful shortcut. For example, if you divide \(P(x)\) by \(x-2\), the remainder is \(P(2)\). You do not need to complete the full long division just to find the remainder.
The factor theorem is a direct consequence of the remainder theorem. It says that \(x-c\) is a factor of \(P(x)\) if and only if \(P(c)=0\). In other words, if the remainder is zero, the divisor is a factor. This connects polynomial division to roots, zeros, and graph intercepts.
Remainder and factor theorem
\[ \text{If } P(x) \text{ is divided by } x-c,\text{ then the remainder is } P(c). \] \[ x-c \text{ is a factor of } P(x) \Longleftrightarrow P(c)=0. \]
For example, let \(P(x)=x^3-8\). If we divide by \(x-2\), the remainder is \(P(2)=2^3-8=0\). Since the remainder is zero, \(x-2\) is a factor of \(x^3-8\). This is why \(x^3-8=(x-2)(x^2+2x+4)\).
Common mistakes in polynomial division
The first common mistake is forgetting missing terms. If a polynomial is \(x^4+3x^2-1\), there is no \(x^3\)-term and no \(x\)-term. When doing long division by hand, it helps to write \(x^4+0x^3+3x^2+0x-1\). The zero placeholders keep the columns aligned. The calculator handles missing terms automatically, but students should still understand why placeholders matter.
The second common mistake is subtracting incorrectly. In polynomial long division, you subtract the entire product of the divisor and the quotient term. This means every sign inside the product changes during subtraction. For example, subtracting \(x^2-3x\) gives \(-x^2+3x\). Many errors come from changing only the first sign and forgetting the rest.
The third common mistake is stopping too early. You should continue division until the degree of the remainder is lower than the degree of the divisor. If the divisor is linear, the remainder must be a constant. If your remainder still has an \(x\)-term, you probably need one more division step.
The fourth common mistake is using synthetic division when it does not apply. Synthetic division is mainly for divisors like \(x-c\). If the divisor is \(x^2+1\) or \(2x-3\), be careful. Long division is more general and safer for mixed cases.
The fifth common mistake is forgetting the remainder in the final answer. If the remainder is not zero, the final rational expression is not just the quotient. It is \(Q(x)+\frac{R(x)}{D(x)}\). Leaving out the remainder changes the value of the expression.
Practice problems
Use these practice problems after trying the calculator. Divide each polynomial and write the quotient and remainder.
- \((x^3-8)\div(x-2)\)
- \((x^3+6x^2+11x+6)\div(x+1)\)
- \((2x^3+3x^2-5x+6)\div(x-2)\)
- \((x^4-1)\div(x^2-1)\)
- \((3x^3+5x^2-4x+7)\div(x+3)\)
- \((2x^4-x^3+6x-5)\div(x^2+2)\)
Answers: \(1)\ Q=x^2+2x+4,\ R=0\). \(2)\ Q=x^2+5x+6,\ R=0\). \(3)\ Q=2x^2+7x+9,\ R=24\). \(4)\ Q=x^2+1,\ R=0\). \(5)\ Q=3x^2-4x+8,\ R=-17\). \(6)\ Q=2x^2-x-4,\ R=8x+3\).
Polynomial division FAQs
What is polynomial division?
Polynomial division is the process of dividing one polynomial by another to find a quotient and a remainder. It follows the identity \(P(x)=D(x)Q(x)+R(x)\).
What is the quotient in polynomial division?
The quotient is the main polynomial result after division. It is the polynomial \(Q(x)\) in the identity \(P(x)=D(x)Q(x)+R(x)\).
What is the remainder in polynomial division?
The remainder is what is left after division stops. Its degree must be lower than the degree of the divisor.
When is a polynomial divisor a factor?
A divisor is a factor when the remainder is zero. For a divisor \(x-c\), this is equivalent to \(P(c)=0\).
What is the difference between long division and synthetic division?
Long division works for general polynomial divisors. Synthetic division is a shortcut mainly used for divisors of the form \(x-c\).
How do I write a polynomial division answer with a remainder?
Write the answer as \(Q(x)+\frac{R(x)}{D(x)}\), where \(Q(x)\) is the quotient, \(R(x)\) is the remainder, and \(D(x)\) is the divisor.
Related Num8ers resources
Polynomial division connects to polynomial multiplication, factoring, rational expressions, the remainder theorem, and polynomial equations. Use these related Num8ers resources to help students continue learning.